Finite Group Databasebeta

← #30564771

Subgroups — #30564771

Order 15616.

2142 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459673C211yesmaximal, characteristic, complemented
2C2×#16840415C211yesmaximal, characteristic, complemented
3#21457746C214yesmaximal, complemented
4#21457739C214yesmaximal, complemented
5#21457544C214yesmaximal, complemented
6#21457539C214yesmaximal, complemented
7#21456744C211yesmaximal, characteristic, complemented
8#21456603C214yesmaximal, complemented
9#21455823C211yesmaximal, characteristic, complemented
10#21455820C211yesmaximal, characteristic, complemented
11#21455042C214yesmaximal, complemented
12#21454842C211yesmaximal, characteristic, complemented
13C61×#3209C211yesmaximal, characteristic, complemented
14C22×#11758226C2212yes—
15C2×#14519800C2218yescomplemented
16#16840497—22no—
17#16840485C2212yescomplemented
18#16840478C2212yescomplemented
19#16840465—22no—
20#16840434C2212yescomplemented
21#16840415—22no—
22#16840415—22no—
23#16840415C2214yes—
24#16840415C2214yes—
25#16840415C2212yescomplemented
26#16840352—22no—
27#16840331—21no—
28#16840312C2211yescharacteristic
29#16840275C2214yes—
30C2×#14519762—22no—
31C2×#14519753C2212yes—
32C2×#14519752—21no—
33#16840242—21no—
34C2×C4×D244C2211yescharacteristic
35C2×C4×D244C2211yescharacteristic
36C2×C4⋊C4×D61—21no—
37#16840136C2212yescomplemented
38#16840112C2212yescomplemented
39#16840102C2218yescomplemented
40#16840090C2212yes—
41#16840076C2212yes—
42#16840049C2212yescomplemented
43#16840043C2212yescomplemented
44#16840000C2218yescomplemented
45#16839994C2218yescomplemented
46#16839964C2214yescomplemented
47#16839964C2212yescomplemented
48#16839958C2212yescomplemented
49#16839923C2218yescomplemented
50#16839923C2218yescomplemented
51#16839912C2212yes—
52#16839910C2218yescomplemented
53#16839901C2214yes—
54#16839898C2212yes—
55C4×#11758201C2214yescomplemented
56C4×D488C2214yescomplemented
57C8×D244C2214yescomplemented
58#16839827C2214yescomplemented
59#16839815C2211yescharacteristic
60#16839757C2218yescomplemented
61C22⋊C8×D61C2214yes—
62C61×#533C2211yescharacteristic
63C2×C4⋊Q8×C61C2211yescharacteristic
64C61×#494C2212yescomplemented
65C61×#487C2212yescomplemented
66C61×#462C2214yescomplemented
67C61×#451C2214yes—
68C61×#431C2211yescharacteristic
69C2×#11758226—24no—
70C2×#11758226—48no—
71C2×#11758226C2318yes—
72C2×#11758226C2314yes—
73C2×#11758222—21no—
74C22×D244—22no—
75C22×D244D411yescharacteristic
76C22×D244C2311yescharacteristic
77C22×C4×D61—21no—
78C22×C4×D61—22no—
79C22×C4×D61C2312yes—
80#14519800—416no—
81C2×#11758201C2314yes—
82C2×#11758201C2318yes—
83C2×D488C2314yes—
84C2×C8×D61C2318yes—
85#14519777C2311yescharacteristic
86#14519774—22no—
87#14519774C2311yescharacteristic
88#14519772—24no—
89#14519766—24no—
90#14519765—22no—
91#14519765—21no—
92#14519762—24no—
93#14519762—24no—
94#14519762—28no—
95#14519762—28no—
96#14519753—24no—
97#14519753C2314yes—
98#14519753C2318yes—
99#14519752—22no—
100#14519752—22no—
101#14519752—24no—
102#14519752—24no—
103#14519752—24no—
104#14519752C2314yes—
105#14519751—21no—
106#14519751C2312yes—
107#14519747—24no—
108C4×Dic122—22no—
109C4×D244—21no—
110C4×D244—21no—
111C4×D244—21no—
112C4×D244—22no—
113C4×D244D412yes—
114C4×D244D412yes—
115C4×D244C2312yes—
116C4×D244C2312yes—
117C4×D244C2312yes—
118C4×D244C2312yes—
119C4×D244C2312yes—
120#14519739—24no—
121#14519737—24no—
122#14519735—22no—
123#14519735C2314yes—
124C2×C4⋊Dic61D411yescharacteristic
125C2×C4⋊Dic61C2311yescharacteristic
126C2×#11758196—22no—
127C4⋊C4×D61—22no—
128C4⋊C4×D61—24no—
129C4⋊C4×D61—24no—
130C2×#11758198—22no—
131C2×#11758198—21no—
132C2×#11758198D412yes—
133#14519711C2314yes—
134#14519708C2314yes—
135C4⋊#92418C2312yes—
136#14519701C2314yes—
137#14519701C2318yes—
138#14519699C2314yes—
139#14519699C2314yes—
140#14519697C2314yes—
141C22⋊#92418C2312yes—
142#14519689C2318yes—
143C4×#92418C2312yes—
144C22×Q8×C61C2312yes—
145C2×Q16×C61C2318yes—
146C4⋊Q8×C61—21no—
147C4⋊Q8×C61—21no—
148C4⋊Q8×C61C2312yes—
149C4⋊Q8×C61C2312yes—
150C4⋊Q8×C61C2311yescharacteristic
151C4⋊Q8×C61C2311yescharacteristic
152#125×C61—21no—
153C22⋊Q8×C61—21no—
154C22⋊Q8×C61C2312yes—
155C4×Q8×C61—21no—
156C4×Q8×C61C2312yes—
157#117×C61—21no—
158C2×C4⋊C4×C61—21no—
159C2×C4⋊C4×C61—21no—
160C2×C42×C61C2311yescharacteristic
161C4⋊C8×C61C2312yes—
162#103×C61C2318yes—
163C22⋊C8×C61C2312yes—
164C4×C8×C61C2312yes—
165C23×D61—21no—
166C23×D61C2×D412yes—
167C2×Q8×C61—22no—
168C2×Q8×C61—21no—
169C2×Q8×C61—44no—
170C2×Q8×C61C2414yes—
171C2×Q8×C61C2412yes—
172C22×C4×C61—22no—
173C22×C4×C61C2×D411yescharacteristic
174C22×C4×C61C2×D411yescharacteristic
175C22×C4×C61C2411yescharacteristic
176C2×C22⋊D61—22no—
177C22×Dic61—21no—
178C22×Dic61C2×D412yes—
179#11758226—216no—
180#11758226—48no—
181#11758226—48no—
182#11758226—416no—
183#11758222—44no—
184C2×D244—24no—
185C2×D244—21no—
186C2×D244—24no—
187C2×D244—28no—
188C2×D244—21no—
189C2×D244—24no—
190C2×D244—24no—
191C2×D244—21no—
192C2×D244—48no—
193C2×D244C2×D412yes—
194C2×D244C2×D412yes—
195C2×D244C2412yes—
196C2×D244C2412yes—
197C2×C4×D61—22no—
198C2×C4×D61—22no—
199C2×C4×D61—22no—
200C2×C4×D61—28no—
201C2×C4×D61—24no—
202C2×C4×D61—44no—
203C2×C4×D61—48no—
204C2×C4×D61—44no—
205C2×C4×D61C2×D414yes—
206C2×C4×D61C2414yes—
207C2×Dic122—21no—
208Q16×C61—48no—
209C2×C8×C61C2414yes—
210C4⋊C4×C61—22no—
211C4⋊C4×C61—22no—
212C4⋊C4×C61—22no—
213C4⋊C4×C61—22no—
214C4⋊C4×C61—21no—
215C4⋊C4×C61—21no—
216C4⋊C4×C61—21no—
217C4⋊C4×C61—22no—
218C4⋊C4×C61—22no—
219C4⋊C4×C61C2412yes—
220C22⋊C4×C61—22no—
221C42×C61—22no—
222C42×C61C2×D411yescharacteristic
223C42×C61C2×D411yescharacteristic
224C42×C61C2411yescharacteristic
225C42×C61C2411yescharacteristic
226#11758201—28no—
227#11758201—48no—
228#11758198—28no—
229#11758198—24no—
230#11758198—24no—
231#11758198—22no—
232#11758198—22no—
233#11758198—22no—
234#11758198—22no—
235#11758198—24no—
236#11758198C2×D414yes—
237#11758198C2×D414yes—
238C4⋊Dic61—21no—
239C4⋊Dic61—21no—
240C4⋊Dic61C2×D412yes—
241C4⋊Dic61C2×D412yes—
242C4⋊Dic61C2412yes—
243C4⋊Dic61C2412yes—
244#11758196—28no—
245#11758196—24no—
246C4×Dic61—24no—
247C2×#92418C2414yes—
248D488—28no—
249C8×D61—48no—
250C23×C61C22×D411yescharacteristic
251C22×D61—24no—
252C22×D61—22no—
253C22×D61—24no—
254C22×D61—22no—
255C22×D61—42no—
256C22×D61—42no—
257C22×D61—42no—
258C22×D61—42no—
259C22×D61C22×D414yes—
260Q8×C61—24no—
261Q8×C61—28no—
262Q8×C61—44no—
263Q8×C61—81no—
264C2×C4×C61—22no—
265C2×C4×C61—22no—
266C2×C4×C61—22no—
267C2×C4×C61—24no—
268C2×C4×C61—41no—
269C2×C4×C61—44no—
270C2×C4×C61#13514yescomplemented
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C22×D411yescharacteristic
275C2×C4×C61C22×D411yescharacteristic
276C2×C4×C61C2511yescharacteristic
277C22⋊D61—44no—
278C2×Dic61—22no—
279C2×Dic61—22no—
280C2×Dic61—42no—
281C2×Dic61—42no—
282C2×Dic61C22×D414yes—
283D244—24no—
284D244—24no—
285D244—24no—
286D244—22no—
287D244—24no—
288D244—416no—
289D244—42no—
290D244—81no—
291D244—84no—
292D244—84no—
293D244H(5,𝔽2)14yes—
294C4×D61—28no—
295C4×D61—44no—
296C4×D61—416no—
297C4×D61—44no—
298C4×D61—44no—
299C4×D61—44no—
300C4×D61—84no—
301Dic122—42no—
302C8×C61—24no—
303#92418—24no—
304#61565—611nomaximal, Sylow
305C22×C61—21no—
306C22×C61—21no—
307C22×C61C2×#13512yes—
308C22×C61C23×D411yescharacteristic
309C2×D61—24no—
310C2×D61—24no—
311C2×D61—24no—
312C2×D61—24no—
313C2×D61—48no—
314C2×D61—42no—
315C2×D61—42no—
316C2×D61—42no—
317C2×D61—42no—
318C2×D61—42no—
319C2×D61—84no—
320C4×C61—21no—
321C4×C61—21no—
322C4×C61—21no—
323C4×C61—42no—
324C4×C61—44no—
325C4×C61—42no—
326C4×C61C2×#13512yes—
327C4×C61C2×#13512yes—
328C4×C61C2×H(5,𝔽2)12yes—
329Dic61—24no—
330Dic61—42no—
331Dic61—42no—
332#3506—611no—
333C2×#547—611no—
334#3303—614no—
335#3302—614no—
336#3284—614no—
337#3283—614no—
338#3209D6111yescharacteristic, complemented
339#3206—611no—
340#3190—614no—
341#3101—611no—
342#3100—611no—
343#3002—614no—
344#2972—611no—
345C2×C61—22no—
346C2×C61#348411yescharacteristic
347C2×C61C22×#13511yescharacteristic
348C2×C61#358611yescharacteristic
349D61—44no—
350D61—44no—
351D61—82no—
352C22×#41—612no—
353C2×#135—618no—
354#559—1222no—
355#556—612no—
356#555—612no—
357#553—1222no—
358#550—612no—
359#547—614no—
360#547—614no—
361#547—612no—
362#547—1222no—
363#547—1222no—
364#538—1222no—
365#535—1221no—
366#533C2×D6111yescharacteristic
367#532—611no—
368C2×C4⋊Q8C2×D6111yescharacteristic
369#525—614no—
370C2×#123—1222no—
371C2×C22⋊Q8—612no—
372C2×#121—1221no—
373#520—1221no—
374C2×C4×D4—611no—
375C2×C4×D4—611no—
376C22×C4⋊C4—1221no—
377#494C2×D6112yescomplemented
378#492—612no—
379#487—612no—
380#487C2×D6112yescomplemented
381#485—618no—
382#483—612no—
383#480—612no—
384#476—612no—
385#474—612no—
386#467—618no—
387#465—618no—
388#462C2×D6114yescomplemented
389#461—612no—
390#461—614no—
391#459—612no—
392#452—618no—
393#452—618no—
394#451C2×D6114yes—
395#450—612no—
396#449—618no—
397#448—614no—
398#447—612no—
399C4×SD16—614no—
400C4×D8—614no—
401C8×D4—614no—
402C8×D4—614no—
403#431—611no—
404#431C2×D6111yescharacteristic
405#416—618no—
406C2×C22⋊C8—614no—
407C61#6156511yescharacteristic, complemented, Sylow
408C2×#41—618no—
409C2×#41—614no—
410C2×#41—1224no—
411C2×#41—1221no—
412C2×#41—2448no—
413C22×Q8C22×D6112yes—
414C22×D4—611no—
415C22×D4—611no—
416C22×D4—1222no—
417C23×C4—612no—
418C23×C4—1222no—
419C23×C4—1221no—
420#135—24416no—
421C2×Q16C22×D6118yes—
422C2×SD16—614no—
423C2×SD16—618no—
424C2×D8—614no—
425C22×C8—618no—
426C4⋊Q8—21no—
427C4⋊Q8—21no—
428C4⋊Q8—611no—
429C4⋊Q8C22×D6112yes—
430C4⋊Q8C22×D6112yes—
431C4⋊Q8C22×D6111yescharacteristic
432C4⋊Q8C22×D6111yescharacteristic
433#127—611no—
434#127—1222no—
435#126—1224no—
436#125—21no—
437#124—1222no—
438#124—1224no—
439#124—1221no—
440#123—1228no—
441#123—1228no—
442#123—1224no—
443#123—1224no—
444C22⋊Q8—21no—
445C22⋊Q8—618no—
446C22⋊Q8—614no—
447C22⋊Q8—1224no—
448C22⋊Q8C22×D6112yes—
449#121—612no—
450#121—614no—
451#121—1224no—
452#121—1224no—
453#121—1224no—
454#121—1222no—
455#121—1222no—
456#121—1221no—
457#120—1224no—
458C4×Q8—21no—
459C4×Q8—1222no—
460C4×Q8C22×D6112yes—
461C4×D4—612no—
462C4×D4—612no—
463C4×D4—612no—
464C4×D4—612no—
465C4×D4—612no—
466C4×D4—612no—
467C4×D4—612no—
468C4×D4—1222no—
469C4×D4—1224no—
470C4×D4—1224no—
471C4×D4—1221no—
472C4×D4—1221no—
473C4×D4—1221no—
474#117—21no—
475#117—614no—
476#117—1222no—
477C2×C4⋊C4—21no—
478C2×C4⋊C4—21no—
479C2×C4⋊C4—611no—
480C2×C4⋊C4—611no—
481C2×C4⋊C4—1224no—
482C2×C4⋊C4—1224no—
483C2×C4⋊C4—1222no—
484C2×C4⋊C4—1222no—
485C2×C22⋊C4—612no—
486C2×C22⋊C4—1222no—
487C2×C22⋊C4—1221no—
488C2×C42C22×D6111yescharacteristic
489#107—614no—
490#106—614no—
491C4⋊C8—612no—
492C4⋊C8C22×D6112yes—
493#103—618no—
494#103—614no—
495#103C22×D6118yes—
496#102—614no—
497#102—614no—
498#102—614no—
499C22⋊C8—612no—
500C22⋊C8—618no—
501C22⋊C8C22×D6112yes—
502C4×C8—612no—
503C4×C8C22×D6112yes—
504C24—612no—
505C24—1221no—
506#41—12216no—
507#41—2448no—
508#41—2448no—
509#41—24416no—
510#41—2444no—
511C2×Q8—22no—
512C2×Q8—21no—
513C2×Q8—44no—
514C2×Q8—1221no—
515C2×Q8C23×D6114yes—
516C2×Q8C23×D6112yes—
517C2×D4—612no—
518C2×D4—612no—
519C2×D4—612no—
520C2×D4—612no—
521C2×D4—1224no—
522C2×D4—1224no—
523C2×D4—1224no—
524C2×D4—1228no—
525C2×D4—1222no—
526C2×D4—1221no—
527C2×D4—1221no—
528C2×D4—1221no—
529C2×D4—1224no—
530C2×D4—2448no—
531C22×C4—22no—
532C22×C4—612no—
533C22×C4—614no—
534C22×C4—614no—
535C22×C4—1224no—
536C22×C4—1228no—
537C22×C4—1221no—
538C22×C4—1222no—
539C22×C4—1222no—
540C22×C4—1222no—
541C22×C4—2444no—
542C22×C4—2448no—
543C22×C4—2444no—
544C22×C4D4×D6111yescharacteristic
545C22×C4D4×D6111yescharacteristic
546C22×C4C23×D6111yescharacteristic
547Q16—48no—
548SD16—1228no—
549SD16—2448no—
550D8—1228no—
551C2×C8—614no—
552C2×C8—2448no—
553C2×C8C23×D6114yes—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—22no—
561C4⋊C4—21no—
562C4⋊C4—22no—
563C4⋊C4—612no—
564C4⋊C4—612no—
565C4⋊C4—612no—
566C4⋊C4—612no—
567C4⋊C4—1228no—
568C4⋊C4—1224no—
569C4⋊C4—1221no—
570C4⋊C4—1221no—
571C4⋊C4C23×D6112yes—
572C22⋊C4—22no—
573C22⋊C4—614no—
574C22⋊C4—614no—
575C22⋊C4—1224no—
576C22⋊C4—1228no—
577C22⋊C4—1224no—
578C22⋊C4—1224no—
579C22⋊C4—1222no—
580C22⋊C4—1222no—
581C22⋊C4—1222no—
582C22⋊C4—1222no—
583C42—22no—
584C42—1224no—
585C42D4×D6111yescharacteristic
586C42D4×D6111yescharacteristic
587C42C23×D6111yescharacteristic
588C42C23×D6111yescharacteristic
589C23—614no—
590C23—1224no—
591C23—1224no—
592C23—1222no—
593C23—1222no—
594C23—2442no—
595C23—2442no—
596C23—2442no—
597C23—2442no—
598C23C2×D4×D6111yescharacteristic
599Q8—24no—
600Q8—28no—
601Q8—44no—
602Q8—81no—
603Q8—2442no—
604D4—614no—
605D4—1224no—
606D4—1224no—
607D4—1224no—
608D4—1222no—
609D4—1224no—
610D4—4881no—
611D4—4884no—
612D4—4884no—
613D4—24416no—
614D4—2444no—
615D4—2442no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—41no—
621C2×C4—44no—
622C2×C4—614no—
623C2×C4—1228no—
624C2×C4—1222no—
625C2×C4—1222no—
626C2×C4—4884no—
627C2×C4—2444no—
628C2×C4—24416no—
629C2×C4—2442no—
630C2×C4—2442no—
631C2×C4—2444no—
632C2×C4—2444no—
633C2×C4—2444no—
634C2×C4#1451980014yescomplemented
635C2×C4C2×D4×D6111yescharacteristic
636C2×C4C2×D4×D6111yescharacteristic
637C2×C4C2×D4×D6111yescharacteristic
638C2×C4C2×D4×D6111yescharacteristic
639C2×C4C2×D4×D6111yescharacteristic
640C2×C4C24×D6111yescharacteristic
641C8—24no—
642C8—1224no—
643C22—21no—
644C22—21no—
645C22—1224no—
646C22—1224no—
647C22—1224no—
648C22—1224no—
649C22—4884no—
650C22—2448no—
651C22—2442no—
652C22—2442no—
653C22—2442no—
654C22—2442no—
655C22—2442no—
656C22C2×#1451980012yes—
657C22C22×D4×D6111yescharacteristic
658C4—21no—
659C4—21no—
660C4—21no—
661C4—42no—
662C4—44no—
663C4—42no—
664C4—1224no—
665C4—2442no—
666C4—2442no—
667C4C2×#1451980012yes—
668C4C2×#1451980012yes—
669C4H(5,𝔽2)×D6112yes—
670C2—22no—
671C2—4882no—
672C2—2444no—
673C2—2444no—
674C2#2145945311yescharacteristic
675C2C22×#1451980011yescharacteristic
676C2#2146060211yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3209C2
socleC22×C61C23×D4
hypercenter#3209D61
hypocenterC61#61565
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21459673C2semidirect
2. C2×#16840415C2semidirect
3. #21457746C2semidirect
4. #21457739C2semidirect
5. #21457544C2semidirect
6. #21457539C2semidirect
7. #21456744C2semidirect
8. #21456603C2semidirect
9. #21455823C2semidirect
10. #21455820C2semidirect
11. #21455042C2semidirect
12. #21454842C2semidirect
13. C61×#3209C2semidirect
15. C2×#14519800C22semidirect
17. #16840485C22semidirect
18. #16840478C22semidirect
20. #16840434C22semidirect
25. #16840415C22semidirect
37. #16840136C22semidirect
38. #16840112C22semidirect
39. #16840102C22semidirect
42. #16840049C22semidirect
43. #16840043C22semidirect
44. #16840000C22semidirect
45. #16839994C22semidirect
46. #16839964C22semidirect
47. #16839964C22semidirect
48. #16839958C22semidirect
49. #16839923C22semidirect
50. #16839923C22semidirect
52. #16839910C22semidirect
55. C4×#11758201C22semidirect
56. C4×D488C22semidirect
57. C8×D244C22semidirect
58. #16839827C22semidirect
60. #16839757C22semidirect
64. C61×#494C22semidirect
65. C61×#487C22semidirect
66. C61×#462C22semidirect
270. C2×C4×C61#135semidirect
338. #3209D61semidirect
377. #494C2×D61semidirect
380. #487C2×D61semidirect
388. #462C2×D61semidirect
407. C61#61565semidirect
634. C2×C4#14519800semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214596732
C2×#168404152
#214577462
#214577392
#214575442
#214575392
#214567442
#214566032
#214558232
#214558202
#214550422
#214548422
C61×#32092
#6156561